Infinite generation of the kernels of the Magnus and Burau representations

Abstract

Consider the kernel ${Mag}_g$ of the Magnus representation of the Torelli group and the kernel ${Bur}_n$ of the Burau representation of the braid group. We prove that for $g\ge 2$ and for $n\ge 6$ the groups ${Mag}_g$ and ${Bur}_n$ have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of “Johnson-type” homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of ${Bur}_n$, we do this with the assistance of a computer calculation.